Analytical models of the wall-pressure spectrum under a turbulent boundary layer with adverse pressure gradient

This paper presents a comprehensive analytical approach to the modelling of wall-pressure fluctuations under a turbulent boundary layer, unifying and expanding the analytical models that have been proposed over many decades. The Poisson equation governing pressure fluctuations is Fourier transformed in the wavenumber domain to obtain a modified Helmholtz equation, which is solved with a Green’s function technique. The source term of the differential equations is composed of turbulence–mean shear and turbulence–turbulence interaction terms, which are modelled separately within the hypothesis of a joint normal probability distribution of the turbulent field. The functional expression of the turbulence statistics is shown to be the most critical point for a correct representation of the wall-pressure spectrum. The effect of various assumptions on the shape of the longitudinal correlation function of turbulence is assessed in the first place with purely analytical considerations using an idealised flow model. Then, the effect of the hypothesis on the spectral distribution of boundary-layer turbulence on the resulting wall-pressure spectrum is compared with the results of direct numerical simulation computations and pressure measurements on a controlled-diffusion aerofoil. The boundary layer developing over the suction side of this aerofoil in test conditions is characterised by an adverse pressure gradient. The final part of the paper discusses the numerical aspect of wall-pressure spectrum computation. A Monte Carlo technique is used for a fast evaluation of the multi-dimensional integral formulation developed in the theoretical part.